If you also own the main domain, it makes no difference to me. If you are sharing hosting with someone else, it increases the number of people I have to trust. In general though, it cost so little to get a separate domain that I'd go for it.

Either you are too cheap to get your own domain/hosting, in which case I don't want you holding or responsible for any of my funds. Or you are too incompetent to set up your own domain/hosting, in which case I don't trust that you can responsibly secure your site against the most novice of hackers.

The only thing I care about in an URL is the https:// part. Using plaintext http looks to me as though the admin is still sleeping in the 90's before security was important.

Apart from that make sure that the site itself looks professional and trustworthy. The current BTC sites look pretty amateurish on average

Bitcoin Core developer [PGP]Warning: For most, coin loss is a larger risk than coin theft. A disk can die any time. Regularly back up your wallet through File → Backup Wallet to an external storage or the (encrypted!) cloud. Use a separate offline wallet for storing larger amounts.